The maybe monad

module Monad.Instance.Maybe.Commutative {o ℓ e} {C : Category o ℓ e} (distributive : Distributive C) where
  open Category C
  open M C
  open MR C
  open MP C
  open HomReasoning
  open Equiv
  open Distributive distributive
  open import Categories.Category.Distributive.Properties distributive
  open Cocartesian cocartesian
  open Cartesian cartesian using (terminal; products)
  open BinaryProducts products renaming (unique to ⟨⟩-unique)
  open Terminal terminal
  open CartesianMonoidal cartesian using (monoidal)

  open import Monad.Instance.Maybe distributive
  open import Monad.Instance.Maybe.Strong distributive

  open Symmetric (symmetric C cartesian) using (braided)

  open Commutative

  distribute₄ : ∀ {A B C D} → (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹ {A + B} {C} {D} ≈ [ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹
  distribute₄ = Iso⇒Epi (IsIso.iso isIsoʳ) ((distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) ([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) (begin 
    (((distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) ∘ distributeʳ)                                                                                                ≈⟨ ∘[] ⟩ 
    [ (((distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) ∘ (i₁ ⁂ id)) , (((distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹) ∘ (i₂ ⁂ id)) ]                           ≈⟨ []-cong₂ (pullʳ ((refl⟩∘⟨ (⁂-cong₂ refl (sym (+-unique id-comm-sym id-comm-sym)))) ○ sym (distributeˡ⁻¹-natural i₁ id id))) (pullʳ (((refl⟩∘⟨ (⁂-cong₂ refl (sym (+-unique id-comm-sym id-comm-sym)))) ○ sym (distributeˡ⁻¹-natural i₂ id id)))) ⟩ 
    [ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ ((i₁ ⁂ id) +₁ (i₁ ⁂ id)) ∘ distributeˡ⁻¹ , (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ ((i₂ ⁂ id) +₁ (i₂ ⁂ id)) ∘ distributeˡ⁻¹ ] ≈⟨ []-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ distributeʳ⁻¹-i₁ distributeʳ⁻¹-i₁)) (pullˡ (+₁∘+₁ ○ +₁-cong₂ distributeʳ⁻¹-i₂ distributeʳ⁻¹-i₂)) ⟩
    [ (i₁ +₁ i₁) ∘ distributeˡ⁻¹ , (i₂ +₁ i₂) ∘ distributeˡ⁻¹ ]                                                                                                       ≈˘⟨ []∘+₁ ⟩ 
    ([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹))                                                                                                      ≈˘⟨ pullʳ (cancelʳ (IsIso.isoˡ isIsoʳ)) ⟩ 
    (([ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹) ∘ distributeʳ)                                                                      ∎)

  commutativeMaybe : Commutative braided maybeStrong
  commutativeMaybe .commutes {X} {Y} = begin 
    [ id , i₂ ] ∘ ((swap +₁ id) ∘ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap +₁ id) ∘ (id +₁ !) ∘ distributeˡ⁻¹                                                                                                ≈⟨ refl⟩∘⟨ (pullˡ (+₁∘+₁ ○ +₁-cong₂ identityʳ identityˡ)) ⟩ 
    [ id , i₂ ] ∘ ((swap +₁ id) ∘ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap +₁ !) ∘ distributeˡ⁻¹                                                                                                             ≈⟨ pullˡ ([]∘+₁ ○ []-cong₂ identityˡ refl) ⟩ 
    [ (swap +₁ id) ∘ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap , i₂ ∘ ! ] ∘ distributeˡ⁻¹                                                                                                                     ≈⟨ ([]-cong₂ (refl⟩∘⟨ (pullʳ distributeˡ⁻¹∘swap)) (refl⟩∘⟨ !-unique (! ∘ distributeʳ⁻¹))) ⟩∘⟨refl ⟩ 
    [ (swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap) ∘ distributeʳ⁻¹ , i₂ ∘ ! ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹                                                                                             ≈˘⟨ pullˡ ([]∘+₁ ○ []-cong₂ assoc²' assoc) ⟩ 
    [ (swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap) , i₂ ∘ ! ] ∘ (distributeʳ⁻¹ +₁ distributeʳ⁻¹) ∘ distributeˡ⁻¹                                                                                          ≈⟨ refl⟩∘⟨ distribute₄ ⟩ 
    [ (swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap) , i₂ ∘ ! ] ∘ [ i₁ +₁ i₁ , i₂ +₁ i₂ ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹                                                                ≈⟨ pullˡ ∘[] ⟩ 
    [ [ (swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap) , i₂ ∘ ! ] ∘ (i₁ +₁ i₁) , [ (swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap) , i₂ ∘ ! ] ∘ (i₂ +₁ i₂) ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹   ≈⟨ ([]-cong₂ []∘+₁ []∘+₁) ⟩∘⟨refl ⟩ 
    [ [ ((swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap)) ∘ i₁ , (i₂ ∘ !) ∘ i₁ ] , [ ((swap +₁ id) ∘ (id +₁ !) ∘ (swap +₁ swap)) ∘ i₂ , (i₂ ∘ !) ∘ i₂ ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ≈⟨ ([]-cong₂ ([]-cong₂ (pullʳ (pullʳ inject₁)) (pullʳ (sym (!-unique (! ∘ i₁))))) ([]-cong₂ (pullʳ (pullʳ inject₂)) (pullʳ (sym (!-unique (! ∘ i₂)))))) ⟩∘⟨refl ⟩ 
    [ [ (swap +₁ id) ∘ (id +₁ !) ∘ i₁ ∘ swap , i₂ ∘ ! ] , [ (swap +₁ id) ∘ (id +₁ !) ∘ i₂ ∘ swap , i₂ ∘ ! ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹                                       ≈⟨ ([]-cong₂ ([]-cong₂ (refl⟩∘⟨ (pullˡ (inject₁ ○ identityʳ))) refl) ([]-cong₂ (refl⟩∘⟨ (pullˡ inject₂)) refl)) ⟩∘⟨refl ⟩ 
    [ [ (swap +₁ id) ∘ i₁ ∘ swap , i₂ ∘ ! ] , [ (swap +₁ id) ∘ (i₂ ∘ !) ∘ swap , i₂ ∘ ! ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹                                                         ≈⟨ ([]-cong₂ ([]-cong₂ (pullˡ inject₁) refl) ([]-cong₂ (pullˡ (pullˡ (inject₂ ○ identityʳ))) refl)) ⟩∘⟨refl ⟩ 
    [ [ (i₁ ∘ swap) ∘ swap , i₂ ∘ ! ] , [ (i₂ ∘ !) ∘ swap , i₂ ∘ ! ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹                                                                              ≈⟨ ([]-cong₂ ([]-cong₂ (pullʳ swap∘swap) refl) ([]-cong₂ (pullʳ (sym (!-unique (! ∘ swap)))) refl)) ⟩∘⟨refl ⟩ 
    [ id +₁ ! , [ i₂ ∘ ! , i₂ ∘ ! ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹                                                                                                               ≈⟨ pullˡ []∘+₁ ⟩ 
    [ (id +₁ !) ∘ distributeˡ⁻¹ , [ i₂ ∘ ! , i₂ ∘ ! ] ∘ distributeˡ⁻¹ ] ∘ distributeʳ⁻¹                                                                                                                ≈⟨ ([]-cong₂ refl ((sym ∘[]) ⟩∘⟨refl)) ⟩∘⟨refl ⟩ 
    [ (id +₁ !) ∘ distributeˡ⁻¹ , (i₂ ∘ [ ! , ! ]) ∘ distributeˡ⁻¹ ] ∘ distributeʳ⁻¹                                                                                                                   ≈⟨ ([]-cong₂ refl (pullʳ (sym (!-unique ([ ! , ! ] ∘ distributeˡ⁻¹))))) ⟩∘⟨refl ⟩ 
    [ (id +₁ !) ∘ distributeˡ⁻¹ , i₂ ∘ ! ] ∘ distributeʳ⁻¹                                                                                                                                             ≈˘⟨ ([]-cong₂ (cancelʳ swap∘swap) (pullʳ (sym (!-unique (! ∘ swap))))) ⟩∘⟨refl ⟩ 
    [ (((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap) ∘ swap , (i₂ ∘ !) ∘ swap ] ∘ distributeʳ⁻¹                                                                                                                  ≈˘⟨ pullˡ []∘+₁ ⟩ 
    [ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap , i₂ ∘ ! ] ∘ (swap +₁ swap) ∘ distributeʳ⁻¹                                                                                                                   ≈˘⟨ pullʳ distributeˡ⁻¹∘swap ⟩ 
    ([ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap , i₂ ∘ ! ] ∘ distributeˡ⁻¹) ∘ swap                                                                                                                           ≈˘⟨ pullˡ (pullˡ ([]∘+₁ ○ []-cong₂ identityˡ (refl⟩∘⟨ identityˡ))) ⟩ 
    [ id , i₂ ] ∘ (((((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap) +₁ id ∘ !) ∘ distributeˡ⁻¹) ∘ swap                                                                                                            ≈˘⟨ refl⟩∘⟨ (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²) ○ pullˡ (pullˡ (+₁∘+₁ ○ +₁-cong₂ identityʳ refl))) ⟩ 
    [ id , i₂ ] ∘ ((id +₁ !) ∘ distributeˡ⁻¹ +₁ id) ∘ (swap +₁ id) ∘ ((id +₁ !) ∘ distributeˡ⁻¹) ∘ swap                                                                                                ∎